Unitary Matrix Research Articles

Commutators of skew-involutions

Let $\mathrm{SL}_n(\mathbb{F})$ be the group of all $n\times n$ matrices over a field $\mathbb{F}$ with determinant $1$. Denote by $I$ ($I_n$) the ($n\times n$) identity matrix. A matrix $A$ is called skew-involution if $A^2=-I$. It is proved that every matrix in $\mathrm{SL}_{2n}(\mathbb{F})$ is a product of at most three commutators of skew-involutions if $\mathbb{F}\ne\mathbb{Z}_3$ and $\mathrm{SL}_{2n}(\mathbb{F})\ne\mathrm{SL}_{2}(\mathbb{Z}_2)$, and at most four commutators of skew-involutions if $\mathbb{F}=\mathbb{Z}_3$ and $n>1$. Every complex symplectic matrix is a product of two commutators of complex symplectic skew-involutions, and every real symplectic matrix is a product of not more than four commutators of real symplectic skew-involutions.

Local certification of unitary operations

In this work, we analyze the local certification of unitary quantum channels, which is a natural extension of quantum hypothesis testing. A particular case of a quantum channel operating on two systems corresponding to product states at the input, is considered. The goal is to minimize the probability of the type II error, given a specified maximum probability of the type I error, considering assistance through entanglement with auxiliary systems. Our result indicates connection of the local certification problem with a product numerical range of unitary matrices. We show that the optimal local strategy does not need usage of auxiliary systems and requires only single round of one-way classical communication. Moreover, we compare local and global certification strategies and show that typically local strategies are optimal, yet in some extremal cases, where global strategies make no errors, local ones may fail miserably. Finally, some application for local certification of von Neumann measurements are discussed as well.

Comprehensive classification of the algebra generated by two idempotent matrices

For two idempotent matrix P,Q∈Cn×n, let alg(In,P,Q) denote the smallest subalgebra of Cn×n that contains P,Q and the identity matrix In. This paper provides a complete classification of alg(In,P,Q) without imposing any restrictions on P and Q. As a result of this classification, the issue of group invertibility within alg(In,P,Q) is fully resolved.

Beyond the Blackwell order in dichotomies

I establish a translation invariance property of the Blackwell order for dichotomies and show that the norm of the distance from the identity matrix may be interpreted as a measure of informativeness. The better experiment is closer to the fully revealing experiment; this measure extends the Blackwell order, is complete, and prior-independent.

Exploring Bounded Component Analysis Using an ℓ∞ Norm Criterion

In this paper we propose a new criterion for the Blind Source Separation (BSS) of antisparse bounded sources, based on the sum of the ℓ∞-norm of the sources. Based on the observation that the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the ℓ∞-norm of the sources, unless it is the identity matrix, the minimization of the sum of the ℓ∞-norm of the sources can be used for the estimation of a separation matrix. To that, a Principle Component Analysis technique followed by a Givens Rotations based optimization method can be used for the separation of independent bounded sources. Also, the Givens Rotations based optimization method can be used for the separation of correlated bounded sources mixed by a rotation matrix. We theoretically analyze the proposed criterion and assess its performance through numerical simulations involving three distinct types of bounded signals. Our theoretical and experimental findings underscore the efficacy of the ℓ∞ norm as a suitable contrast function for antisparse bounded sources, showcasing its superior performance relative to a state-of-the-art algorithm.

The Relationship Between Institutional Pedagogy and the Brand Identity Formation of Higher Education Institutions

This study sought to facilitate insight into the potential role of pedagogy in the brand identity formation of higher education institutions (HEIs) through a study of selected HEIs in The Gambia. Specifically, the study sought to address the following research question: What role do pedagogical practices play in building the brand identity of selected universities in The Gambia? The intra-paradigm qualitative mixed method of data collection underpinned the research design. This research design facilitated a preliminary analysis of the contents of institutional documents and social media postings. This process was followed by telephonic and virtually mediated in-depth interviews through which this researcher explored the interactionist interpretations, recollections, experiences, and opinions of 54 participants (students and staff) on the themes of institutional brand management practices, institutional pedagogical practices, institutional brand identity, and the links between pedagogical practices and institutional brand identity. The study used the Corporate Brand Identity Matrix as a supporting analysis framework. The findings show a relationship between pedagogical practices and institutional brand identity formation. The evidence suggests that the selected HEIs use hardly differentiated production-style portfolios of academic courses to pursue largely unengaged students, prospective students, and other stakeholders. Further findings indicate that teaching and learning practice is dominated by academic staff's discretionary use of transmissive pedagogy. This insight emerged against the background of additional evidence, which shows a link between pedagogy policy and practices of HEIs and stakeholder impressions. A synthesis of these findings culminated in the emergence of the pedagogy-based higher education brand identity matrix (P-HEBIM), which this study proposes as a novel framework for the branding of HEIs. The study sets out a practitioner guide on how higher education managers can pursue the institutional brand management priorities of branding strategy development, competitor intelligence, and brand communication using the P-HEBIM as a framework.

Structural Analysis of Virus Regulatory N6-Methyladenosine (m6A) Machinery of the Black Flying Fox (Pteropus alecto) and the Egyptian Fruit Bat (Rousettus aegyptiacus) Shows Evolutionary Conservation Amongst Mammals

Background: N6-methyladenosine (m6A) is an abundant RNA epitranscriptomic modification in eukaryotes. The m6A machinery includes cellular writer, eraser and reader proteins that regulate m6A. Pteropus alecto (P. alecto) (the Australian black flying fox) and Rousettus aegyptiacus (R. aegyptiacus) (the Egyptian fruit bat) are bats associated with several viral zoonoses yet neglected in the field of m6A epigenetics studies. Objectives: This study utilises various bioinformatics and in silico tools to genetically identify, characterise and annotate the m6A machinery in P. alecto and R. aegyptiacus. Methods: A range of bioinformatic tools were deployed to comprehensively characterise all known m6A-associated proteins of P. alecto and R. aegyptiacus. Results: Phylogenetically, the m6A fat mass and obesity-associated protein (FTO) eraser placed the order Chiroptera (an order including all bat species) in a separate clade. Additionally, it showed the lowest identity matrices in P. alecto and R. aegyptiacus when compared to other mammals (74.1% and 72.8%) and Homo sapiens (84.0% and 76.1%), respectively. When compared to humans, genetic loci-based analysis of P. alecto and R. aegyptiacus showed syntenic conservation in multiple flanking genes of 8 out the 10 m6A-associated genes. Furthermore, amino acid alignment and protein tertiary structure of the two bats’ m6A machinery demonstrated conservation in the writers but not in erasers and readers, compared to humans. Conclusions: These studies provide foundational annotation and genetic characterisation of m6A machinery in two important species of bats which can be exploited to study bat–virus interactions at the interface of epitranscriptomics.

Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions

For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.

Structural Changes in Copper Slags During Slow Cooling

The objects of the study were converter slags from the Balkhash copper plant in their initial state and after heat treatment. Using mineralogical and X-ray phase analysis, scanning electron microscopy (SEM), and electron probe microanalysis (EPMA), it was found that the initial converter slag and its thermally treated samples have identical matrices with almost complete coincidence in mineral and phase compositions. The distinguishing feature is the quantitative ratio of mineral components in the slag mass. Almost all of the iron is oxidized and present in the form of fayalite, magnetite, and magnetite, with other elements (silicon, copper, zinc, and aluminum) incorporated into its lattice. The structure of all slag samples indicates an association of sulfur exclusively with copper. Copper in the slags was identified in both metallic and sulfide forms. Slow cooling of the converter slag after its remelting contributes to the reduction in the sulfide–metal suspension in the volume of the melt and its coarsening. During slow cooling, structural changes occur not only in the main oxide part of the slag but also in the polymetallic globules.

Elimination and Factorization

Summary If a matrix A has rank r, then its row echelon form (from elimination) contains the identity matrix in its first r independent columns. How do we intepret the matrix F that appears in the remaining columns of that echelon form? The matrix F multiplies those first r independent columns of A to give its n–r dependent columns. Then F reveals bases for the row space and the nullspace of the original matrix A. And F is the key to the column-row factorization A = CR .

Oxidation Potential of 2,6-Dimethyl-1,4-dihydropyridine Derivatives Estimated by Structure Descriptors

Linear relationships, expressing the electrochemical properties of molecules as functions of structure, give insight into the associated electrochemical process and are a tool for prediction. Many biological activities rely on water-based dissociation, making electrochemical properties a bridge between structure and activity. Motivated by a previous study, a replica is made here on a different dataset in order to validate/invalidate the previously reported results. There are several methods for obtaining structure-based descriptors. Some of the methods have been devised to account for molecular topology, some to account for molecular geometry, and others to account for both. Two methods are involved here to derive structure-based descriptors and further obtain structure–property relationships (FMPI and ChPE). In order to express structure descriptors, both FMPI and ChPE express first the topology of the molecule, using the heavy atoms identity matrix and the heavy atoms adjacency matrix, both square symmetric matrices in the belief that symmetry is one major factor of molecular stability. A set of 2,6-dimethyl-1,4-dihydropyridine derivatives with oxidation peak potentials and coulometrically determined number of electrons experimental data is subjected to the search for structure–activity relationships. Even if the 2,6-dimethyl-1,4-dihydropyridine is a symmetric compound (of Cs point group), their derivatives are generally not symmetric (9 out of 24 are asymmetric). The dataset is subjected to descriptive and inferential statistics in order to filter out the most relevant structure–activity relationship. The geometry is built using three levels of theory (one from molecular mechanics and two others from density functionals, of which one accounts for the interaction with water as solvent). One challenge of picking one out of two reported measured values is dealt with by calculating the likelihood associated with the two choices. Relevant structure–activity models are extracted and discussed. The use of in vivo (in water, SM8 model) models in geometry optimization (from MMFF94 and B3LYP and to M06 + Water SM8) results in a precision gain, but this is, in most of the cases, not statistically significant, and this can be considered a negative result.

Disentangled contrastive learning for fair graph representations

Graph Neural Networks (GNNs) play a key role in efficiently learning node representations of graph-structured data through message passing, but their predictions are often correlated with sensitive attributes and thus lead to potential discrimination against some groups. Given the increasingly widespread applications of GNNs, solutions are required urgently to prevent algorithmic discrimination associated with GNNs, to protect the rights of vulnerable groups and to build trustworthy artificial intelligence. To learn the fair node representations of graphs, we propose a novel framework, the Fair Disentangled Graph Neural Network (FDGNN). With the proposed FDGNN framework, we enhance data diversity by generating instances that have identical sensitivity values but different adjacency matrices through data augmentation. Additionally, we design a counterfactual augmentation strategy for constructing instances with varying sensitive values while preserving the same adjacency matrices, thereby balancing the distribution of sensitive values across different groups. Subsequently, we employ a disentangled contrastive learning strategy to acquire disentangled representations of non-sensitive attributes such that sensitive information does not affect the prediction of node information. Finally, the learned fair representations of non-sensitive attributes are employed for building a fair predictive model. Extensive experiments on three real-world datasets demonstrate that FDGNN yields the best fairness predictions compared to the baseline methods. Additionally, the results demonstrate the potential of disentanglement in learning fair representations.

Symmetric Cyclic Quantum Teleportation using Channel Parameter Matrix

Abstract The Channel Parameter Matrix (CPM) is a matrix containing general variables used to formulate the channel used for transmitting quantum bit (qubit) in quantum teleportation. CPM is an important parameter to describe the whole process in quantum teleportation. The scheme used in this process is symmetric cyclic. In cyclic scheme, there is three parties: Alice, Bob, and Charlie. For symmetric cyclic scheme, Alice sends one qubit to Bob, Bob sends one qubit to Charlie, Charlie sends one qubit to Alice. Then the size of CPM for symmetric is 8 x 8. Further derivation show that CPM depend on normalization constant, unitary matrix, and type of measurement.

One-step ahead short-term hourly global solar radiation forecasting with a dynamical system based on classification of days

The data-driven method is a common approach to solar irradiation prediction. However, its weakness is that it is highly dependent on the characteristics of the prior data, which leads to a lag delay in forecasting. Therefore, this paper proposes a prediction model that considers data-driven, data-cycle variations, and statistical information for solving such problems. Firstly, the periodicity of the data is removed using the Fourier transform. Secondly, monthly daily characterization matrices are constructed to predict from a global perspective without including information from previous time series. Finally, the coupled autoregressive and dynamical system (CARDS) model is improved using the identity matrix. Comparing the model proposed in this paper with seven other popular prediction models, the error can be reduced by 30.2 % on average at four locations in the northern and southern hemispheres. Moreover, the proposed model does not increase the computational burden of existing models.

Constraining non-unitary neutrino mixing using matter effects in atmospheric neutrinos at INO-ICAL

The mass-induced neutrino oscillation is a well established phenomenon that is based on the unitary mixing among three light active neutrinos. Remarkable precision on neutrino mixing parameters over the last decade or so has opened up the prospects for testing the possible non-unitarity of the standard 3ν mixing matrix, which may arise in the seesaw extensions of the Standard Model due to the admixture of three light active neutrinos with heavy isosinglet neutrinos. Because of this non-unitary neutrino mixing (NUNM), the oscillation probabilities among the three active neutrinos would be altered as compared to the probabilities obtained assuming a unitary 3ν mixing matrix. In such a NUNM scenario, neutrinos can experience an additional matter effect due to the neutral current interactions with the ambient neutrons. Atmospheric neutrinos having access to a wide range of energies and baselines can experience a significant modifications in Earth’s matter effect due to NUNM. In this paper, we study in detail how the NUNM parameter α32 affects the muon neutrino and antineutrino survival probabilities in a different way. Then, we place a comparable and complementary constraint on α32 in a model independent fashion using the proposed 50 kt magnetized Iron Calorimeter (ICAL) detector under the India-based Neutrino Observatory (INO) project, which can efficiently detect the atmospheric νμ and ν¯μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\overline{\ u}}_{\\mu } $$\\end{document} separately in the multi-GeV energy range. Further, we discuss the advantage of charge identification capability of ICAL and the impact of uncertainties in oscillation parameters while constraining α32. We also compare the α32 sensitivity of ICAL with that of future long-baseline experiments DUNE and T2HK in isolation and combination.

A Generic Analysis of Nucleon Decay Branching Fractions in Flipped SU(5) Grand Unification

In flipped SU(5) grand unified theories, the partial decay lifetimes of certain nucleon decay channels depend generically on an unknown unitary matrix, which arises when left-handed lepton fields are embedded into anti-fundamental representations of SU(5). This dependency is particularly relevant when the neutrino mass matrix has a generic structure, introducing uncertainty in the prediction of nucleon decay branching fractions within flipped SU(5). In this paper, we demonstrate that this uncertainty can be parametrized using two parameters, which can be determined by measuring the partial lifetimes of p→π0e+, p→π0μ+, and n→π0ν¯. In addition, we establish upper limits on the ratios of the decay widths of these channels, offering a potential method to test flipped SU(5) in future nucleon decay experiments.

NsDCC: dual-level contrastive clustering with nonuniform sampling for scRNA-seq data analysis.

Dimensionality reduction and clustering are crucial tasks in single-cell RNA sequencing (scRNA-seq) data analysis, treated independently in the current process, hindering their mutual benefits. The latest methods jointly optimize these tasks through deep clustering. However, contrastive learning, with powerful representation capability, can bridge the gap that common deep clustering methods face, which requires pre-defined cluster centers. Therefore, a dual-level contrastive clustering method with nonuniform sampling (nsDCC) is proposed for scRNA-seq data analysis. Dual-level contrastive clustering, which combines instance-level contrast and cluster-level contrast, jointly optimizes dimensionality reduction and clustering. Multi-positive contrastive learning and unit matrix constraint are introduced in instance- and cluster-level contrast, respectively. Furthermore, the attention mechanism is introduced to capture inter-cellular information, which is beneficial for clustering. The nsDCC focuses on important samples at category boundaries and in minority categories by the proposed nearest boundary sparsest density weight assignment algorithm, making it capable of capturing comprehensive characteristics against imbalanced datasets. Experimental results show that nsDCC outperforms the six other state-of-the-art methods on both real and simulated scRNA-seq data, validating its performance on dimensionality reduction and clustering of scRNA-seq data, especially for imbalanced data. Simulation experiments demonstrate that nsDCC is insensitive to "dropout events" in scRNA-seq. Finally, cluster differential expressed gene analysis confirms the meaningfulness of results from nsDCC. In summary, nsDCC is a new way of analyzing and understanding scRNA-seq data.

Theoretical error estimates for computing the matrix logarithm by Padé-type approximants

In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss–Legendre quadrature rules. These formulas can be interpreted as Padé approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the identity matrix, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced.

Variational approach to learning photonic unitary operators

Structured light, light tailored in its internal degrees of freedom, has become topical in numerous quantum and classical information processing protocols. In this work, we harness the high dimensional nature of structured light modulated in the transverse spatial degree of freedom to realize an adaptable scheme for learning unitary operations. Our approach borrows from concepts in variational quantum computing, where a search or optimization problem is mapped onto the task of finding a minimum ground state energy for a given energy/goal function. We achieve this by a pseudo-random walk procedure over the parameter space of the unitary operation, implemented with optical matrix-vector multiplication enacted on arrays of Gaussian modes by exploiting the partial Fourier transforming capabilities of a cylindrical lens in the transverse degree of freedom for the measurement. We outline the concept theoretically, and experimentally demonstrate that we are able to learn optical unitary matrices for dimensions d = 2, 4, 8, and 16 with average fidelities of >90%. Our work advances high dimensional information processing and can be adapted to both process and quantum state tomography of unknown states and channels.

Normalization of ZZ instanton amplitudes in type 0B minimal superstring theory

Abstract We study ZZ instanton corrections in the (2, 4k) $$ \mathcal{N} $$ N = 1 minimal superstring theory with the type 0B GSO projection, which becomes the type 0B $$ \mathcal{N} $$ N = 1 super-JT gravity in the k → ∞ limit. Each member of the (2, 4k) family of theories has two phases distinguished by the sign of the Liouville bulk cosmological constant. The worldsheet method for computing the one-loop normalization constant multiplying the instanton corrections gives an ill-defined answer in both phases. We fix these divergences using insights from string field theory and find finite, unambiguous results. Each member of the (2, 4k) family of theories is dual to a double-scaled one-matrix integral, where the double-scaling limit can be obtained starting either from a unitary matrix integral with a leading one-cut saddle point, or from a hermitian matrix integral with a leading two-cut saddle point. The matrix integral exhibits a gap-closing transition, which is the same as the double-scaled Gross-Witten-Wadia transition when k = 1. We also compute instanton corrections in the double-scaled matrix integral for all k and in both phases, and find perfect agreement with the string theory results.